Given a scalar function defined on a meshed surface, its level set component tree can be computed by a fast algorithm. This tree structure allows for an adaptation to meshes of the Maximally Stable Extremal Regions (MSER) method. Applied to the mesh curvature, this algorithm extracts significant curvature level lines and segments 3D surfaces into smooth parts separated by curves with high curvature. Segmentation results are shown on high resolution meshes of archaeological and industrial pieces. They compare favorably with MSER segmentations of pictures of the same objects.